We present measurements of the velocity power spectrum and constraints on the growth rate of structure f sigma(8), at redshift zero, using the peculiar motions of 2062 galaxies in the completed 2MASS Tully-Fisher survey (2MTF). To accomplish this we introduce a model for fitting the velocity power spectrum including the effects of non-linear redshift space distortions (RSD), allowing us to recover unbiased fits down to scales k= 0.2 h Mpc(-1) without the need to smooth or grid the data. Our fitting methods are validated using a set of simulated 2MTF surveys. Using these simulations we also identify that the Gaussian distributed estimator for peculiar velocities of Watkins & Feldman is suitable for measuring the velocity power spectrum, but sub-optimal for the 2MTF data compared to using magnitude fluctuations delta m, and that, whilst our fits are robust to a change in fiducial cosmology, future peculiar velocity surveys with more constraining power may have to marginalize over this. We obtain scale-dependent constraints on the growth rate of structure in two bins, finding f sigma(8) = [0.55(-0.13)(+0.16), 0.40(-0.17)(+0.16)] in the ranges k = [0.007-0.055, 0.55-0.150] h Mpc(-)1. We also find consistent results using four bins. Assuming scale-independence we find a value f sigma(8) = 0.51(-0.08)(+0.09), a similar to 16 per cent measurement of the growth rate. Performing a consistency check of general relativity (GR) and combining our results with cosmic microwave background data only we find gamma = 0.45(-0.11)(+0.10), a remarkable constraint considering the small number of galaxies. All of our results are completely independent of the effects of galaxy bias, and fully consistent with the predictions of GR (scale-independent f sigma(8) and gamma approximate to 0.55).